group theory

# Contents

## Idea

What is called the cosmic Galois group is a motivic Galois group that naturally acts on structures in renormalization in quantum field theory. The actual renormalization group is a 1-parameter subgroup of the cosmic Galois group.

In more detail, in (Connes-Marcolli 04) the authors consider a differential equation satisfied by divergences appearing in the Hopf algebra formulation of renormalization (see there). This leads to a category of “equisingular flat connections” that turns out to be a Tannakian category, meaning that it is equivalent to the category of modules over some pro-algebraic group $G$. The authors observe that $G$ acts on any renormalizable theory in a nice way. Due to this property Pierre Cartier referred to this as the cosmic Galois group:

La parenté de plus en plus manifeste entre le groupe de Grothendieck-Teichmüller d’une part, et le groupe de renormalisation de la Théorie Quantique des Champs n’est sans doute que la première manifestation d’un groupe de symétrie des constantes fondamentales de la physique, une espèce de groupe de Galois cosmique!“ (Pierre Cartier according to Connes 12, see also the end of Cartier 01).

This cosmic Galois group $G$ is (non-canonically) isomorphic to some motivic Galois group.

In (Kitchloo-Morava 12) the cosmic Galois group is related to the motivic/Tannakian group of a motivic stabilization of the symplectic category of symplectic manifolds and Lagrangian correspondences between them, the stable symplectic category (Kitchloo 12).

## References

The “cosmic Galois group” in renormalization theory was introduced in

The name originates in

• Pierre Cartier A mad day’s work: from Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry, Bull. Amer. Math. Soc. 38 (2001), 389-408, pdf

A similar statement about a motivic Galois group acting on the space of deformation quantizations of a free field theory appeared earlier in

Review and exposition includes

A technical review of aspects of this is in